English
If each A_i carries an algebra structure over R via S_i, then the product ∏ i A_i carries a natural Algebra structure over the product ∏ i S_i, defined componentwise with suitable compatibility conditions.
Русский
Если каждый A_i имеет структуру алгебры над R через S_i, то произведение ∏ i A_i естественно становится Algebra над ∏ i S_i, по координатам.
LaTeX
$$$ \\text{Algebra } (\\prod_i S_i) ((\\prod_i A_i)) $$$
Lean4
instance [∀ i, Algebra (S i) (A i)] : Algebra (Π i, S i) (Π i, A i)
where
algebraMap := Pi.ringHom fun _ ↦ (algebraMap _ _).comp (Pi.evalRingHom S _)
commutes' _ _ := funext fun _ ↦ Algebra.commutes _ _
smul_def' _ _ := funext fun _ ↦ Algebra.smul_def _ _
